OpenRevise

Exam questions · Maths · Number Without a Calculator

Powers, Roots and Indices

  • 7 exam questions
  • 16 marks
  • 10 quick checks
  1. 1 Work out [2 marks]

    Work out (a) \(11^2\) [1 mark] (b) \(\sqrt[3]{125}\) [1 mark]

    Show answerHide answer

    Model answer

    (a) \(11 \times 11 = 121\). (b) \(5 \times 5 \times 5 = 125\), so \(\sqrt[3]{125} = 5\).

    Mark scheme

    • (a) 121 — B1
    • (b) 5 — B1
  2. 2 Simplify [2 marks]

    Simplify (a) \(x^4 \times x^5\) [1 mark] (b) \(y^8 \div y^3\) [1 mark]

    Show answerHide answer

    Model answer

    (a) Add the indices: \(x^{4+5} = x^9\). (b) Subtract the indices: \(y^{8-3} = y^5\).

    Mark scheme

    • (a) \(x^9\) — B1
    • (b) \(y^5\) — B1
  3. 3 Work out [2 marks]

    Work out the value of \(2^{-3} \times 4\). Give your answer as a fraction.

    Show answerHide answer

    Model answer

    \(2^{-3} = \dfrac{1}{8}\), so \(\dfrac{1}{8} \times 4 = \dfrac{4}{8} = \dfrac{1}{2}\). Alternatively, \(4 = 2^2\), so \(2^{-3} \times 2^2 = 2^{-1} = \dfrac{1}{2}\).

    Mark scheme

    • \(2^{-3} = \dfrac{1}{8}\) or \(2^{-3} \times 2^2\) — M1
    • \(\dfrac{1}{2}\) — A1
  4. 4 Find [3 marks]

    (a) Find the value of \(n\) when \(2^n = \dfrac{1}{32}\). [2 marks] (b) Find the value of \(n\) when \(3^n \times 3^4 = 3\). [1 mark]

    Show answerHide answer

    Model answer

    (a) \(\dfrac{1}{32} = \dfrac{1}{2^5} = 2^{-5}\), so \(n = -5\). (b) \(n + 4 = 1\), so \(n = -3\).

    Mark scheme

    • (a) \(32 = 2^5\) or \(\dfrac{1}{2^5}\) — M1
    • (a) \(-5\) — A1
    • (b) \(-3\) — B1
  5. 5 Work out [3 marks]

    Work out the value of (a) \(125^{\frac{1}{3}}\) [1 mark] (b) \(81^{\frac{3}{4}}\) [2 marks]

    Show answerHide answer

    Model answer

    (a) The cube root of 125 is 5. (b) \(81^{\frac{1}{4}} = 3\), then \(3^3 = 27\).

    Mark scheme

    • (a) 5 — B1
    • (b) \(\sqrt[4]{81} = 3\) or \(81^{\frac{1}{4}} = 3\) — M1
    • (b) 27 — A1
  6. 6 Show that [2 marks]

    Show that \(5^7 + 5^7 + 5^7 + 5^7 + 5^7 = 5^8\).

    Show answerHide answer

    Model answer

    There are five lots of \(5^7\), so the sum is \(5 \times 5^7 = 5^1 \times 5^7 = 5^{1+7} = 5^8\).

    Mark scheme

    • \(5 \times 5^7\) — M1
    • \(5^1 \times 5^7 = 5^8\) with a clear conclusion — A1
  7. 7 Simplify [2 marks]

    Simplify \((3a^2b^3)^2\).

    Show answerHide answer

    Model answer

    Square every part: \(3^2 \times (a^2)^2 \times (b^3)^2 = 9a^4b^6\).

    Mark scheme

    • Two of \(9\), \(a^4\), \(b^6\) correct — M1
    • \(9a^4b^6\) — A1

Quick check

  1. 1

    Between which two whole numbers does \(\sqrt{40}\) lie?

    1. A6 and 7
    2. B5 and 6
    3. C4 and 5
    4. D7 and 8
    Show answerHide answer

    A: 6 and 7

    \(6^2 = 36\) and \(7^2 = 49\), and 40 is between 36 and 49.

  2. 2

    Write \(16 \times 8\) as a single power of 2.

    1. A\(4^7\)
    2. B\(2^{24}\)
    3. C\(2^{12}\)
    4. D\(2^7\)
    Show answerHide answer

    D: \(2^7\)

    \(16 = 2^4\) and \(8 = 2^3\), so \(2^4 \times 2^3 = 2^7\).

  3. 3

    What is the value of \(\sqrt{196}\)?

    1. A98
    2. B14
    3. C16
    4. D13
    Show answerHide answer

    B: 14

    14 × 14 = 196.

  4. 4

    What is the value of \(4^3\)?

    1. A12
    2. B81
    3. C64
    4. D256
    Show answerHide answer

    C: 64

    \(4 \times 4 \times 4 = 64\), not \(4 \times 3 = 12\).

  5. 5

    Simplify \(a^5 \times a^3\).

    1. A\(a^2\)
    2. B\(a^{15}\)
    3. C\(a^8\)
    4. D\(2a^8\)
    Show answerHide answer

    C: \(a^8\)

    Add the indices when multiplying: \(a^{5+3} = a^8\).

  6. 6

    Simplify \((3^2)^4\).

    1. A\(3^8\)
    2. B\(9^8\)
    3. C\(3^6\)
    4. D\(3^{16}\)
    Show answerHide answer

    A: \(3^8\)

    For a power of a power, multiply the indices: \(3^{2 \times 4} = 3^8\).

  7. 7

    What is the value of \(5^{-2}\)?

    1. A\(-10\)
    2. B\(\dfrac{1}{25}\)
    3. C\(-25\)
    4. D\(\dfrac{1}{10}\)
    Show answerHide answer

    B: \(\dfrac{1}{25}\)

    A negative index means a reciprocal: \(5^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25}\).

  8. 8

    What is the value of \(7^0\)?

    1. A1
    2. BUndefined
    3. C0
    4. D7
    Show answerHide answer

    A: 1

    Any non-zero number to the power 0 is 1.

  9. 9

    Simplify \(2^6 \div 2^{-2}\).

    1. A\(2^4\)
    2. B\(2^{-3}\)
    3. C\(2^8\)
    4. D\(2^{-12}\)
    Show answerHide answer

    C: \(2^8\)

    Subtract the indices: \(6 - (-2) = 8\), so the answer is \(2^8\).

  10. 10

    What is the value of \(8^{\frac{1}{3}}\)?

    1. A24
    2. B\(\dfrac{8}{3}\)
    3. C\(\dfrac{1}{8}\)
    4. D2
    Show answerHide answer

    D: 2

    A power of one third means the cube root, and \(2 \times 2 \times 2 = 8\).